Fix any moisture problems before cleaning the wood. If the root problem has not been addressed, the wood will just get moldy again.
Open windows if possible to ensure that your workspace is well-ventilated. If the mold problem is severe and widespread put on a face mask.
Don rubber gloves and scrub away surface mold with hot soapy water. It is important to remove as much of the mold as possible, not just kill it with bleach. Scrub brushes or Scotch-Brite pads are more effective than washcloths, but be careful not to scrub the finish off the wood.
Rinse the wood thoroughly with clear water and dry it well.
Mix nine parts water with one part bleach and apply it directly to the wood. Allow the solution to remain on the wood for at least twelve hours before rinsing it off.